5^21 x 4^11 = 2 x 10^n

5^21 * 2^22 = 2 * 2^n * 5^n
5^21 * 2^22 = 2^n+1 * 5^n
Since bases are same you can equate the exponents
Hence n = 21

How did you go from 5^21 * 4^11

to

5^21 * 2^22?

Putting in my calculator I can see that 4^11 = 2^22 but I'm not sure why as it doesn't work with all numbers. For instance....ahhh...never mind. I was trying to divide the first number by 2, then multiply the exponent by 2 but that obviously wasn't working. Then I realized 2 was the square root of 4 and tried some other numbers. For instance, 9^11 = 3^22.

So you can take the square root of the base and double the exponent and the value is the same? I didn't know you could do that. I also would never have thought of taking 2 * 2^n and making it 2^n+1.

Thanks for the explanation!

Even after the explanation this one was hard so I want to write it out for anyone else struggling with it:

5^21 * 4^11 = 2 * 10^n

5^21 * 2^22 = 2 * 2^n * 5^n

5^21 * 2^22 = 2^n+1 * 5^n

2^n+1 / 2^22 * 5^n / 5^21 = 1

2^n+1-22 * 5^n-21 = 1

2^n-21 * 5^n-21 = 1

10^n-21 = 1

Now, since we know that anything raised to the zero power is 1 we can say

10^0 = 1 therefore

10^n-21 = 10^0 = 1 therefore

n-21 = 0

n = 21

sorry but cybermusing seems easier. try breaking any given number into prime numbers. they tend to show a much more clearer relation

5^21 * 4^11 = 2 * 10^n
=> 5^21 * (2^2)^11 = 2 * 10^n
=> 5^21 * 2^22 = 2 * 10^n
=> 2 * 10^21 = 2 * 10^n
equating n on both sides
we have n=21